Chapter 4: Research Design

Testing a hypothesis

Cross tabulation

Assuming we have a categorical independent variable (IV) and a categorical dependent variable (DV):

iv dv
HIGH No
HIGH No
LOW No
HIGH Yes
HIGH Yes
HIGH Yes
HIGH Yes
LOW Yes
LOW Yes
LOW Yes

Cross tabulation: Step 1

Start by calculating the number of observations with each value of each category:

iv dv
HIGH No
HIGH No
LOW No
HIGH Yes
HIGH Yes
HIGH Yes
HIGH Yes
LOW Yes
LOW Yes
LOW Yes
iv
dv LOW HIGH
No 1 2
Yes 3 4
Total 4 6

Cross tabulation: Step 2

Then, calculate the proportion/percentage of observations among each value of the IV.

If the independent variable is in the columns, then the columns should sum to 100%.

If the independent variable is in the rows, then the rows should sum to 100%.

iv
dv LOW HIGH
No 1 2
Yes 3 4
Total 4 6
iv
dv LOW HIGH
No 1 (25%) 2 (33%)
Yes 3 (75%) 4 (67%)
Total 4 6

Cross tabulation: interpretation

Look at what happens to the DV at different values of the IV. If your variables are ordinal, you should be able to identify a direction of the effect.

The proportion of “Yes” values decreases as the IV goes from lower to higher, so this is a negative or inverse relationship.

iv
dv LOW HIGH
No 1 (25%) 2 (33%)
Yes 3 (75%) 4 (67%)
Total 4 6

Using a bar graph or line graph can make these relationships easier to spot:

Cross tabulation: notes

  • Key rule: always calculate percentages or proportions by categories of the independent variable.

    • This allows you to compare groups that are different sizes.
  • If one or both variables are interval-level, you can bin them in order to use them in a cross tab. For instance, you could separate an interval like into a series of age ranges.

Cross tabulation: example

Hypothesis: in a comparison of individuals, independents are less likely to turn out to vote compared to people who support one party or another.

How should I calculate proportions here?

Voter Turnout in 2020 by party ID
Party ID
turnout2020 Democrat Independent Republican
0. Did not vote 335 316 382
1. Voted 3160 560 2714

Cross tabulation: example

Are these results generally consistent with my hypothesis?

Voter Turnout in 2020 by party ID
Party ID
turnout2020 Democrat Independent Republican
0. Did not vote 335 (10%) 316 (36%) 382 (12%)
1. Voted 3160 (90%) 560 (64%) 2714 (88%)

If we think of party ID as an ordered variable, this is a curvilinear relationship.

Row/Column percentages

What happens if I calculate % among the values of the DV?

Here’s the relationship between education and voter turnout with % calculated on education level:

Voter Turnout in 2020 by highest level of education completed
Education
turnout2020 1. Less than high school credential 2. High school credential 3. Some post-high school, no bachelor's degree 4. Bachelor's degree 5. Graduate degree
0. Did not vote 130 (41%) 286 (24%) 380 (15%) 135 (7%) 91 (6%)
1. Voted 185 (59%) 883 (76%) 2148 (85%) 1749 (93%) 1388 (94%)
Note:
Column % in parentheses

The results suggest a positive or direct relationship: as education increases, so does the % turnout.

Row/Column percentages

What happens if I calculate % among the values of the DV?

Here’s the relationship between education and voter turnout with % calculated across voter turnout

Voter Turnout in 2020 by highest level of education completed
Education
turnout2020 1. Less than high school credential 2. High school credential 3. Some post-high school, no bachelor's degree 4. Bachelor's degree 5. Graduate degree
0. Did not vote 130 (13%) 286 (28%) 380 (37%) 135 (13%) 91 (9%)
1. Voted 185 (3%) 883 (14%) 2148 (34%) 1749 (28%) 1388 (22%)
Note:
Row % in parentheses

Here, the results can give the misleading impression that there’s a curvilinear relationship: turnout drops off for Bachelor’s Degrees and above.

Row/Column percentages

Either of these tables might be a valid way to look at these data, but they answer slightly different questions:

  • If I want to compare turnout at different levels of education, then I need to calculate % turnout among people with different levels of education.

  • If I want to compare education among voters and non-voters, then I need to calculate % education among people who voted and didn’t vote.

  • Which variable is the IV or DV is sometimes a theoretical question, but in this case its unlikely that voting is causing people to become more educated, so it probably doesn’t make sense to calculate percentages by voting vs. non-voting.

Mean Comparison

When we have interval level outcome and a categorical independent variable, we can group each observation by values of the IV and then calculate the mean across each group.

For instance I want to examine the relationship between national wealth and carbon emissions. My hypothesis is that wealthier nations will have more emissions compared to poorer nations.

country gdp.percap.5cat co2.percap
Afghanistan 1. $3k or less 0.281803
Albania 3. $10k to $25k 1.936486
Algeria 3. $10k to $25k 3.988271
Angola 2. $3k to $10k 1.194668
Argentina 3. $10k to $25k 3.995881
Armenia 3. $10k to $25k 2.030401
Australia 5. $45k or more 16.308205
Austria 5. $45k or more 7.648816
Azerbaijan 3. $10k to $25k 3.962984
Bahrain 5. $45k or more 20.934996

Mean Comparison

GDP data has been grouped into five categories, so now I just need to calculate the average of CO2 emissions within each group of the ordinal IV:

GDP Per capita range CO2 emissions per capita
1. $3k or less 0.3128312
2. $3k to $10k 1.2680574
3. $10k to $25k 4.4065669
4. $25k to $45k 8.0307610
5. $45k or more 12.3134306

Is this generally consistent with expectations?

Mean Comparison

Here again, the relationship can be easier to conceptualize if we plot it.

Curvilinear Relationships

A relationship like this will rarely be perfectly straight, so “linearity” and “curvilinearity” are partly a matter of degree, but there are some cases where there is a clear “U” shape to the relationship:

iv dv
1. Extremely liberal 6.314
2. Liberal 5.685
3. Slightly liberal 5.001
4. Moderate; middle of the road 4.651
5. Slightly conservative 4.636
6. Conservative 4.974
7. Extremely conservative 5.363

Research Design

Rival Explanations

  • How can we distinguish correlation from causation?

  • This process inevitably requires us to consider rival explanations for an observed relationship:

    • For instance: if I find that social media use correlated with a lower likelihood of turning out to vote, I might ask whether age is a confounder that could explain this correlation.

Confounding

What I want to show is that Fox News viewership is cases a decreased chance of getting a Covid vaccine.

mrdag X Fox News Y Covid Vaccine X->Y

Confounding

There’s a correlation, but I’m concerned this relationship is spurious because I know that things like existing political views are already correlated with media consumption, and those might explain any correlation I see here:

mrdag Z Conservatism X Fox News Z->X Y Covid Vaccine Z->Y

Its possible that this difference in ideology accounts for the entire observed correlation between media habits and vaccines. I can’t really rule this possibility out without further investigation.

Confounding

What if I could randomly assign people to watch Fox News? Random assignment would ensure that nothing is correlated with Fox news viewership.

mrdag Z Conservatism Y Covid Vaccine Z->Y X Fox News X->Y

Ideology may still matter for getting a vaccine, but since if conservatism is randomly distributed between social media users and non-users, it no longer confounds the observed relationship.

Experiments

  • Experiments use random assignment to account for rival explanations. If you randomly assign people to receive a “treatment”, then you can ensure that there is no confounding because nothing is correlated with your IV.

  • The classic examples are in medicine:

    • Group A is randomly assigned to receive a placebo (the control group)

    • Group B is randomly assigned to receive a medicine (the treatment group)

    • After a certain period of time, we compare the outcomes for both groups.

    • Differences between the groups can be attributed to the effect of the treatment (+/- some random sampling error)

Experiments

  • Experiments are considered a “gold standard” because they can account for all kinds of confounding, including confounding caused by unobserved or unexpected relationships.

  • However, they have two key limitations:

    • External validity: results in the lab may not easily translate to results in real life.

    • Feasibility: many interesting questions just can’t be randomly assigned. We can assign “democracy” or “war” or “religion” to people.

Experiments: Field Experiments

Field experiments can lessen the external validity problem by using random assignment in the field.

For instance, one common way to study GOTV messaging is to randomly select households to receive mailers:

Civic duty treatment

Hawthorne treatment

Neighbors treatment Self treatment

From: GERBER, A. S., GREEN, D. P., & LARIMER, C. W. (2008). Social Pressure and Voter Turnout: Evidence from a Large-Scale Field Experiment. American Political Science Review, 102(1), 33–48. doi:10.1017/S000305540808009X

Experiments: Field Experiments

From: GERBER, A. S., GREEN, D. P., & LARIMER, C. W. (2008). Social Pressure and Voter Turnout: Evidence from a Large-Scale Field Experiment. American Political Science Review, 102(1), 33–48. doi:10.1017/S000305540808009X

Experiments: Natural Experiments

Field experiments can face fewer external validity problems, but some things still can’t be experimentally manipulated.

Natural experiments use “quasi” randomization or “randomization by nature” where treatments are assigned more-or-less randomly.

Experiments: Natural Experiments

  • Viewing Fox News isn’t random, but areas where Fox News is lower in the channel order will have more viewers.

  • Channel order is essentially randomly assigned.

  • So, using channel order as a “treatment” assignment might theoretically allow us to account for confounding in an observational setting.

Experiments: Natural Experiments

Other sources of quasi randomization include:

  • Lotteries (like the Vietnam Draft, or the literal lottery)

  • Arbitrary cutoffs (barely winning an election vs. barely losing)

  • Natural disasters and weather events

Still, natural experiments require a mixture of creativity and luck. They’re not available for most questions.

Observational Research

Observational research doesn’t allow us to easily account for rival explanations. So how can we cope?